General many-dimensional excited random walks Mikhail Menshikov - - PowerPoint PPT Presentation

“Simple” ERW in dimension 2 Generalized ERW Proofs

General many-dimensional excited random walks

Mikhail Menshikov Serguei Popov Alejandro Ramirez Marina Vachkovskaia

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

“Simple” ERW in dimension 2 Generalized ERW Proofs

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Excited Random Walk (ERW) was introduced by Benjamini and Wilson (2003). It is a discrete-time process that lives in Zd, and can be informally described as follows:

◮ fix a parameter p ∈ ( 1 2, 1] ◮ if the walk is at a site x which was already visited, it jumps

with probabilities 1/(2d) to the nearest neighbor sites of x

◮ if the process visits a site x for the first time, it jumps to the

right (i.e., in the direction of the first coordinate vector e1) with probability p/d, to the left with probability (1 − p)/d and to the other nearest neighbor sites of x with probability 1/(2d).

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Informal interpretation:

◮ initially each site contains one cookie ◮ the particle eats all cookies it finds ◮ immediately after eating a cookie, the particle gets a “bias”

to the right

◮ no cookie = no bias

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Known results (Benjamini, Wilson, Kozma, Bérard, Ramirez, van ver Hofstad, Holmes), case d ≥ 2:

◮ ERW is transient to the right ◮ ERW is ballistic to the right ◮ LLN ◮ CLT ◮ monotonicity in p in high dimensions

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Proof (main ideas, d = 2):

◮ coupling of ERW X with SRW Y, such that (Xn − Yn) · e1 is

nondecreasing in n and (Xn − Yn) · e2 = 0

◮ tan points for SRW.

A tan point in dimension d = 2 is defined as any site x ∈ Z2 with the property that the ray {x + ke1 : k ≥ 0} is visited by the SRW for the first time at site x.

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

x y

x is a tan point, y is not a tan point

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

It is known (Bousquet-Mélou, Schaeffer (2002)) that with “large” probability, the number of tan points up to time n is at least n

3 4−ε.

Let Rn be the set of sites visited up to time n. |Rn| ≥ the number of tan points of Y by time n (using the coupling with SRW) So (taking ε < 1

4)

|Rn| > n

3 4 −ε ≫ n 1 2

with “large” probability.

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Proof of transience: Mn = Xn − 2p−1

2

|Rn| is a martingale. Azuma inequality: if {Zn}n∈N is a martingale with respect to some filtration, and such that |Zk − Zk−1| < c a.s., then P[|Zn − Z0| ≥ a] ≤ 2 exp

• −

a2 2nc2

• .

So (take a = n

1 2 +δ), since we should have |Rn| ≫ n 1 2 , it holds

also that (again, with “large” probability) Xn ≫ n

1 2 , and then one

can use Borel-Cantelli to obtain transience to the right.

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Proofs of LLN and CLT (here ℓ = e1):

ℓ τ1 τ2 τ3

regeneration structure + estimates on tails of τk+1 − τk

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

What if we modify the model?

◮ drift in cookies not parallel to e1 ◮ different drifts in different cookies ◮ SRW → some RW with zero drift and bounded jumps ◮ etc.

— there are difficulties, because we cannot use the coupling with SRW and tan points!

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

“Simple” ERW in dimension 2 Generalized ERW Proofs

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Generalized ERW is a discrete-time process X in Zd, d ≥ 2, satisfying the following conditions: Condition B. There exists a constant K > 0 such that supn≥0 Xn+1 − Xn ≤ K a.s. Condition E. Let ℓ ∈ Sd−1. We say that Condition E is satisfied with respect to ℓ if there exist h, r > 0 such that for all n P[(Xn+1 − Xn) · ℓ > r | Fn] ≥ h and for all ℓ′ with ℓ′ = 1, on {E(Xn+1 − Xn | Fn) = 0} P[(Xn+1 − Xn) · ℓ′ > r | Fn] ≥ h.

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Condition C+. Let ℓ ∈ Sd−1. We say that Condition C+ is satisfied with respect to ℓ if there exist a λ > 0 such that E(Xn+1 − Xn | Fn) = 0 on {∃ k < n such that Xk = Xn}, and E(Xn+1 − Xn | Fn) · ℓ ≥ λ on {Xk = Xn for all k < n}.

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Results:

Theorem

Let d ≥ 2 and ℓ ∈ Sd−1. Assume that X is a generalized excited random walk in direction ℓ. Then, there exists v = v(d, K, h, r, λ) > 0 such that lim inf

n→∞

Xn · ℓ n ≥ v a.s. Also, for “homogeneous” ERW and ERW in i.i.d. random environment we prove LLN and (averaged) CLT. They follow from the regeneration times argument, using the estimates obtained in the course of the proof of the above result.

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

“Simple” ERW in dimension 2 Generalized ERW Proofs

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Define H(a, b) = {x ∈ Zd : x · ℓ ∈ [a, b]}, and Ln(m) :=

n

• j=0

1{Xj · ℓ ∈ [m, m + 1)}.

Lemma

Let X ′ be a submartingale in direction ℓ with uniformly bounded jumps and uniform ellipticity. Then, for any δ > 0 there exists a constant γ′

1 such that for all m we have

P[Ln(m) ≥ n

1 2 +2δ] ≤ e−γ′ 1nδ. Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Proof (for m = −1):

◮ Optional Stopping Theorem ⇒ with probability ≥ n− 1

2 −δ, X ′

hits H(n

1 2+δ, +∞) before coming back to H(−∞, 0)

◮ Azuma inequality ⇒ no return to H(−∞, 0) after n

additional steps

◮ so, X ′ k · ℓ > 0 for all k ≤ n with probability at least O(n− 1

2 −δ)

◮ analogously, each return to H(−1, 0) will be the last one up

to time n with probability at least O(n− 1

2 −δ)

◮ thus, no more than n

1 2+2δ returns with large probability. Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Lemma

Let Y be a d-dimensional martingale, with uniformly bounded jumps and uniform ellipticity. Then, there exist γ > 0, b ∈ (0, 1) such that, by time m, Y will visit at least m1− b

2 different sites

with probability ≥ 1 − e−mγ. Key fact (d ≥ 2): there exist b ∈ (0, 1) close enough to 1 and γ′

2 > 0 (depending only on K, h, r) such that

E(Yn+1b | Fn) ≥ Ynb1{Yn > γ′

2}.

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Proof:

◮ Optional Stopping Theorem ⇒ starting from 0, Y will

reach N (without coming back to 0) with probability ≥ N−b

◮ Azuma inequality ⇒ no return to 0 after additional N2

jumps

◮ then (use N = m1/2), the probability of not returning to 0

after m steps is at least m−b/2

◮ so, up to time m no more than mb/2 visits to 0 (and to any

• ther point)

◮ thus, have to visit ≥ m1−b/2 different sites.

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Recall that Rn is the set of sites visited up to time n.

Proposition

There exist positive constants α, γ1, γ2 which depend only on d, K, h, r, such that P[|Rn| < n

1 2+α] < e−γ1nγ2

for all n ≥ 1.

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Proof: fix a ∈ (0, 1

2) and ε > 0 in such a way that

(1 − a + ε) ∧ 1 2 + a 2(1 − b) − 4ε

• > 1

2. Consider (for fixed n) the event G =

• |Rn| ≥ 1

2n(1−a+ε)∧( 1

2+ a 2 (1−b)−4ε)

. We need to prove that P[G] ≥ 1 − e−C1nε/2. (1)

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Let Hj := H

• 2(j − 1)n

a 2 , 2(j + 1)n a 2

.

• Hj

Hj+1 ℓ n

a 2

The strip Hj is a trap if |Rn ∩ Hj| ≥ na(1− b

2 )−2ε.

If x is not in a trap, then the particle will hit al least one “new point” in time na−ε with probability ≥ 1 − e−nγ′′ (when the particle walks on previously visited sites, it has zero drift).

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Let us introduce the event G1 = {Ln(k) ≤ n

1 2 +ε for all k ∈ [−Kn, Kn]}.

Let ˆ Lj be the total number of visits to Hj. On {|Rn| < n

1 2 + a 2 (1−b)−4ε} the number of traps is at most

2n

1 2− a 2 −2ε.

On the event G1, we can write

• j

ˆ Lj1{Hj is a trap} ≤ 4n

a 2 × 2n 1 2 − a 2 −2ε × n 1 2+ε = 8n1−ε < n

2, so the total time spent in non-traps is at least n

2.

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Denote σ0 := 0, and, inductively (B(x, r) is a ball with center x and radius r) σk+1 = min

• j ≥ σk + ⌊na−ε⌋ : |Rj ∩ B(Xj, na/2)| ≤ na(1− b

2 )−2ε

. Consider the event G2 =

• at least one new point is hit on each
• f the time intervals [σj−1, σj), j = 1, . . . , 1

2n1−a+ε .

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

The total time spent in non-traps is at least n

2.

On the other hand, up to the moment σk we can have at most kna−ε instances j such that |Rj ∩ B(Xj, na/2)| ≤ na(1− b

2 )−2ε.

So, on the event

j

ˆ Lj1{Hj is a trap} ≤ 8n1−ε we have that σ 1

2n1−a+ε < n.

But then, on the event G2 we have that |Rn| ≥ 1

2n(1−a+ε).

This means that (G1 ∩ G2) ⊂ G, and (1) follows.

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs

Then, after obtaining the last proposition, we can

◮ observe that ℓ · n−1 k=0 E(Xk+1 − Xk | Fk) ≥ λ|Rn| ◮ apply the Azuma inequality to the martingale

Zn = Xn −

n−1

• k=0

E(Xk+1 − Xk | Fk) to obtain displacement estimates (and transience, with Borel-Cantelli)

◮ use regenerations times to obtain the ballisticity (and,

eventually, LLN+CLT).

Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks